Ratner’s theorem takes place on a homogeneous space. Informally, a homogeneous space is a space X which looks “the same” when viewed from any point on that space. For instance, a sphere is a homogeneous space, but the surface of a cube is not (the cube looks different when viewed from a corner than from a point on an edge or on a face). More formally, a homogeneous space is a space X equipped with an action of a group G of symmetries which is transitive: given any two points x, y on the space, there is at least one symmetry g that moves x to y, thus y=gx. (For instance the cube has several symmetries, but not enough to be transitive; in contrast, the sphere has the transitive action of the special orthogonal group SO(3) as its symmetry group.) It is not hard to see that a homogeneous space X can always be identified (as a set with an action of G) with a quotient , where is a subgroup of G; indeed, one can take to be the stabiliser of an arbitrarily chosen point x in X, and then identify with . For instance, the sphere has an obvious action of the special orthogonal group SO(3), and the stabiliser of (say) the north pole can be identified with SO(2), so that the sphere can be identified with SO(3)/SO(2). More generally, any Riemannian manifold of constant curvature is a homogeneous space; for instance, an m-dimensional torus can be identified with , while a surface X of constant negative curvature can be identified with for some subgroup of (e.g. the hyperbolic plane is isomorphic to ). Furthermore, the cosphere bundle of X – the space of unit (co)tangent vectors on X – is also a homogeneous space with structure group . (For instance, the cosphere bundle of the hyperbolic plane is isomorphic to .)

For the purposes of Ratner’s theorem, we only consider homogeneous spaces X in which the symmetry group G is a connected finite-dimensional Lie group, and X is finite volume (or more precisely, it has a finite non-trivial G-invariant measure). Every compact homogeneous space is finite volume, but not conversely; for instance the modular curve is finite volume but not compact (it has a cusp). (The modular curve has two real dimensions, but just one complex dimension, hence the term “curve”; rather confusingly, it is also referred to as the “modular surface”. As for the term “modular”, observe that the moduli space of unimodular lattices in has an obvious action of , with the stabiliser of being , as well as an obvious left action of and so this moduli space can be identified with the modular curve.)

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